Multilevel Methods for PDE-Constrained Optimization Based on Adaptive Discretizations and Reduced Order Models

Stefan Ulbrich* and Stefanie Bott

In this talk we discuss recent developments for multilevel optimization methods. In particular, we propose a multilevel optimization approach that generates a hierarchy of adaptive discretizations during the optimization iteration using adaptive finite-element approximations and reduced order models such as POD. The adaptive refinement strategy is based on a posteriori error estimators for the PDE-constraint, the adjoint equation and the criticality measure. The resulting optimization method allows to use existing adaptive PDE-solvers and error estimators in a modular way. By combining Moreau-Yosida regularization techniques with the multilevel approach the method is able to handle state constraints. We demonstrate the efficiency of the approach by numerical examples.

Mathematics Subject Classification: 49M37 90C30

Keywords: PDE-constrained optimization; adaptivity; reduced order models; multilevel methods

Minisymposion: Adaptivity and Model Order Reduction in PDE Constrained Optimization